Stresses in a Shaft

When equal and opposite torques are applied to a shaft at either end, the shaft undergoes twisting or torsion. This creates internal shearing forces within the shaft's cross-section that resist the applied torque. The magnitude and distribution of these shearing forces depend on the internal torque and the shaft's geometry, establishing a fundamental relationship between applied torque and internal stress.

By analyzing the free-body diagram of a shaft segment and applying equilibrium equations, engineers can show that internal shearing forces correlate directly with internal torque. When shearing force is expressed as the product of shearing stress and area element, substituting this relation yields an expression for torque in terms of shearing stress. This relation must be satisfied by shearing stresses in any shaft cross-section.

Static equilibrium equations establish the overall relationship between torque and shearing stress but do not reveal how stresses distribute across a shaft's cross-section. The distribution of shearing stresses in an elastic shaft is indeterminate by statics alone and requires deformation analysis to determine accurately. This analysis examines how the shaft physically deforms under torsion to establish stress patterns.

The perpendicular distance from the shaft's axis to the shearing force, denoted as r, is critical in calculating the internal torque. This distance determines the moment arm for shearing forces acting on the cross-section. The relationship between torque, shearing stress, and this perpendicular distance is fundamental to understanding how torsional loads are resisted throughout the shaft.

By isolating a segment of the shaft between two cross-sections and drawing its free-body diagram, engineers can visualize the shearing forces exerted by one portion on the adjacent portion. Applying equilibrium conditions to this isolated segment reveals how internal shearing forces develop and relate to the applied torque. This method systematically connects external loads to internal stress states.

The relationship between torque and shearing stress establishes a constraint that must be satisfied in any shaft cross-section, forming the basis for stress analysis. However, this relationship alone does not specify how stresses vary within the cross-section, requiring additional analysis to predict stress distribution. Understanding this limitation is essential for accurate design of circular shaft stresses in linear range applications.

Deformation analysis bridges the gap between the overall torque-stress relationship and actual stress distribution across a shaft's cross-section. While equilibrium equations establish that stresses must satisfy a specific torque relationship, they cannot determine individual stress values at different locations. Analyzing how the shaft physically deforms under torsion reveals the actual stress distribution pattern needed for safe design.